Properties

Label 2-2300-460.367-c0-0-12
Degree $2$
Conductor $2300$
Sign $0.0433 + 0.999i$
Analytic cond. $1.14784$
Root an. cond. $1.07137$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.996 + 0.0871i)2-s + (−0.245 − 0.245i)3-s + (0.984 − 0.173i)4-s + (0.266 + 0.223i)6-s + (−0.965 + 0.258i)8-s − 0.879i·9-s + (−0.284 − 0.199i)12-s + (0.483 − 0.483i)13-s + (0.939 − 0.342i)16-s + (0.0766 + 0.876i)18-s + (−0.707 − 0.707i)23-s + (0.300 + 0.173i)24-s + (−0.439 + 0.524i)26-s + (−0.461 + 0.461i)27-s − 1.53i·29-s + ⋯
L(s)  = 1  + (−0.996 + 0.0871i)2-s + (−0.245 − 0.245i)3-s + (0.984 − 0.173i)4-s + (0.266 + 0.223i)6-s + (−0.965 + 0.258i)8-s − 0.879i·9-s + (−0.284 − 0.199i)12-s + (0.483 − 0.483i)13-s + (0.939 − 0.342i)16-s + (0.0766 + 0.876i)18-s + (−0.707 − 0.707i)23-s + (0.300 + 0.173i)24-s + (−0.439 + 0.524i)26-s + (−0.461 + 0.461i)27-s − 1.53i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0433 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0433 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.0433 + 0.999i$
Analytic conductor: \(1.14784\)
Root analytic conductor: \(1.07137\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (2207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :0),\ 0.0433 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6072040741\)
\(L(\frac12)\) \(\approx\) \(0.6072040741\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.996 - 0.0871i)T \)
5 \( 1 \)
23 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.245 + 0.245i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.483 + 0.483i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + 1.53iT - T^{2} \)
31 \( 1 - 1.28iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 0.347T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1.32 + 1.32i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 0.684iT - T^{2} \)
73 \( 1 + (-0.909 + 0.909i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.913793868736273920701972185351, −8.341721286230426358269018226585, −7.54957237419189692005702254589, −6.69244778321824352010590274752, −6.16832113650709522412646457532, −5.36952171260605009114236072219, −3.98208323369326813589554201440, −3.02416006872553566960028572912, −1.86465280944926769373811120494, −0.61298936351725551699821623431, 1.43474082933483041156028930447, 2.41411548206070951653866499792, 3.54490522584897093398420149650, 4.58667485244162896154441268182, 5.67986244351444526920241356047, 6.30029649671561332681871019699, 7.37582845912079931081019969664, 7.81627163173566183540956419860, 8.714171507954126672136390666204, 9.371101623417726323848680382233

Graph of the $Z$-function along the critical line